1. Model eigenvalues#
Eigenvalues play a crucial role in the dynamics of models, encapsulating the adjustment process within a model. They are pivotal in dynamic models as their sign and magnitude dictate whether a system will stabilize, diverge after a disturbance, or if such a disturbance will induce oscillations.
For macromodels, which are systems of (non)linear difference equations, the initial step involves linearizing the model by computing Jacobi matrices. Since the model may have lags longer than 1 so the linear model can be of an order higher than one, it is transformed into a first-order system by creating a companion matrix. The stability, marginal stability, or instability of the system is assessed based on the eigenvalues of this companion matrix.
The Modelflow library provides a comprehensive set of tools for calculating a model’s eigenvalues, facilitating the assurance of the model’s dynamic stability and identifying potential sources of instability.
1.1. Load the model#
mpak,bline = model.modelload('..\models\pak.pcim',
alfa=0.8,run=True,keep= 'Baseline',start=2023, end=2100)
mpak.basedf = bline
file read: C:\modelflow manual\papers\mfbook\content\models\pak.pcim
1.2. Make some scenarios#
To provide insight into how shocks impact the model results each scenario shocks the oilprice in 2025. The impulse response function then prowides insigth in the dynamic response of the model.
for oilshock in [0.01 ,5. ,10. ]:
oilshockdf = bline.upd(f'<2025> WLDFCRUDE_PETRO + {oilshock}')
_ = mpak(oilshockdf,2023,2100,keep=f'${oilshock} increase in oil prices 2025',alfa=0.7)
1.3. .eigenvalues_plot() Display polar diagram of eigenvalues#
This is a method to get an overview of the eigenvalues. The .eigenvalues_plot() method first finds all eigenvalues in current time frame, then plots a polar diagram of the eigenvalues for selected years.
Inspection of the polar diagrams reveals indicates that most of the eigenvalues are less than one, and that those that have imaginary roots (those that appear off of the 0 degree line), only a few are outside of the unit circle, indicating exploding osiliation. However not all years have such values. which could be a source of model uncertainty (these are hard to see in the figure, but are identified below). However the real values outside the unit circle indicates that the model will have exploding features - more on that later.
Note
Calculation the eigenvalues and corrosponding eigenvectors entails a number of steps:
finding expressions for all partial derivatives with respect to endogeneous variables
creating a model to calculate the partial derivatives.
calculate all partial derivatives for all year
use the calculated partial derivatives to construct a companion matrix for all year
calculate the eigenvalues and eigenvectors.
In modelflow these steps are performed in a special class newton_diff- as is was constructed to enable solving the model using the newton method. When eigenvalues has been calculated once for a model the results are stores in
a instance of this class .stability_newtonand the calculations are not repeated every time the eigenvalues are used.
fig = mpak.eigenvalues_plot(periode=[2023,2024,2099,2100]);
Note
fig can be exportet to file with:
fig.savefile(‘polar.pdf’)
Inspection of the polar diagrams reveals indicates for Pakistan model:
Real eigenvalues larger than 1, suggesting an exploding response to shocks.
There are instances where the eigenvalues indicate a response that is both exploding and oscillating.
Nevertheless, for the explosion and oscillation effects to be significant, they need to persist over several years, which means the occasional occurrences might not have substantial effects.
1.4. the method .eigenvalues_show() Interactive exploration of eigenvalues and vectors#
To enable closer analysis the .eigenvalues_show() function provides a interactive interface to the analysis of models eigenvalues and vectors. It will generate and display a polar plot of eigenvalues and their corresponding eigenvectors
for a selected year.
The user can interact with the plot via a dropdown for year selection, a slider for eigenvalue
selection, and a button to toggle additional plot details. The polar plot dynamically updates
to reflect the selected eigenvalue, displaying its magnitude and phase. Additional information
about the selected eigenvector is also displayed, facilitating a detailed temporal analysis
of the eigenvalues.
by pushing the open plot widget button a widget to look at the impulse response for the variables which
contributes with weights in the eigenvector corosponding to the selected eigenvalue.
mpak.eigenvalues_show()
The plot widget
The impulse response for the one variable in the first Eigenvector
1.5. Impulse response#
The impulse response can be visualized with the .keep_plot function.
Below, observe the impulse response for selected variables. Initially, we present the response for the maximum projection period to illustrate long-term behavior, followed by a shorter period to highlight oscillations in the short run.
For many variables, the response amplifies over the long run, consistent with real eigenvalues above 1. Similarly, some variables show a diminishing oscillation, corresponding with the presence of eigenvalues with imaginary components.
mpak.keep_plot(' PAKNYGDPMKTPCN PAKBNCABFUNDCN PAKGGDBTTOTLCN PAKCCEMISCO2TKN',
diff=True,start=2023,end=2100,samefig=1,legend=1,title='Impact full projection period');
In interpreting this, it’s crucial to acknowledge that the local price level in the baseline grows significantly. Observing the percentage difference from the baseline reveals that measured in this way the response does not explode. So the real impact of the impulse don’t explode.
mpak.keep_plot(' PAKNYGDPMKTPCN PAKBNCABFUNDCN PAKGGDBTTOTLCN PAKCCEMISCO2TKN',
diffpct=True,start=2023,end=2100,samefig=1,legend=1,title='Impact full projection period');
mpak.keep_plot(' PAKNYGDPMKTPCN PAKBNCABFUNDCN PAKGGDBTTOTLCN PAKCCEMISCO2TKN'
,diff=True,start=2023,end=2040,samefig=1,legend=1,title='Impact start of projection period');
